Untitled

Plots for example with Insulated Ends

> with(plots):

Fourier series of the even periodic extension of sin(x) on [0,Pi].

> plot(2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*0)/(1-4*k^2),k=1..30),x=-3*Pi..3*Pi,color=plum,thickness=2,numpoints=1000,title="Even Extension");

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Integral for coefficients.

> int(sin(x)*cos(n*x),x=0..Pi);

-(1+cos(Pi*n))/(-1+n^2)

Plots for various times.

> plot([2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*0)/(1-4*k^2),k=1..30),2/Pi],x=-0..Pi,thickness=2,view=0..1.3,title="t=0.0");

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> plot([2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*0.01)/(1-4*k^2),k=1..30),2/Pi],x=-0..Pi,thickness=2,view=0..1.3,title="t=0.01");

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> plot([2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*0.02)/(1-4*k^2),k=1..30),2/Pi],x=-0..Pi,thickness=2,view=0..1.3,title="t=0.02");

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> plot([2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*0.03)/(1-4*k^2),k=1..30),2/Pi],
x=0..Pi,thickness=2,view=0..1.3,title="t=0.03");

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> plot([2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*0.05)/(1-4*k^2),k=1..30),2/Pi],
x=0..Pi,thickness=2,view=0..1.3,title="t=0.05");

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> plot([2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*0.1)/(1-4*k^2),k=1..30),2/Pi],
x=0..Pi,thickness=2,view=0..1.3,title="t=0.1");

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> plot([2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*0.2)/(1-4*k^2),k=1..30),2/Pi],
x=0..Pi,thickness=2,view=0..1.3,title="t=0.2");

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> plot([2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*0.3)/(1-4*k^2),k=1..30),2/Pi],
x=0..Pi,thickness=2,view=0..1.3,title="t=0.3");

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> plot([2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*0.6)/(1-4*k^2),k=1..30),2/Pi],
x=0..Pi,thickness=2,view=0..1.3,title="t=0.6");

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> plot([2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*1.0)/(1-4*k^2),k=1..30),2/Pi],
x=0..Pi,thickness=2,view=0..1.3,title="t=1.0");

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>

>

The animation.

> a:=animate(2/Pi + 4/Pi * sum(cos(2*k*x)*exp(-4*k^2*t)/(1-4*k^2),k=1..30),
x=0..Pi,
t=0..1,thickness=2,view=0..1.3,title="Animation for Insultated Ends Example"):

> ave:=plot(2/Pi,x=0..Pi,color=green,thickness=2,view=0..1.3):

> display(a,ave);

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>